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Wednesday, November 25, 2015

Has CCSS Affected Instruction

Brookings, an outfit that is usually a reliable provider of pro-reform clue-free baloney, offers an interesting question from non-resident senior fellow Tom Loveless: Has Common Core influenced instruction?

It's a worthy question. We've talked a lot about how CCSS has affected policy and evaluation and assessment, but has it actually affected what teachers do in the classroom?

The proponents of the Core never developed a way to answer that question because their assertion has always been that we would see the effects on instruction in the flowering of a million awesome test scores. But the 2015 NAEP scores turned out to be a big bowl of proofless pudding, and so now we're left to ask whether the Common Core tree fell in the classroom forest without making a sound, or if it never fell at all.

In the rush to argue whether CCSS has positively or negatively affected American education, these speculations are vague as to how the standards boosted or depressed learning.

In other words, Core fans are unable to get any more specific than their original thoughts that Common Core Standards would somehow magically infuse classrooms, leading to super-duper test scores. Of course, they also assumed that teachers were blithering incompetently in their classrooms and that adhering to awesome standards would mean a change. Loveless notes a 2011 survey in which 77% of teachers said they thought the new math standards were the same as their old math standards. So there's one vote for, "No, the standards changed nothing."

Then Loveless drops this wry observation:

For teachers, the novelty of CCSS should be dissipating.

Yes, the "novelty" is surely fading away. I could jump to the conclusion that Loveless is one more deeply clueless Brookings guy, but he follows that up with these lines:

Common Core’s advocates placed great faith in professional development to implement the standards. Well, there’s been a lot of it. Over the past few years, millions of teacher-hours have been devoted to CCSS training. Whether all that activity had a lasting impact is questionable.

Loveless cites some research that tells us what we mostly know-- after a new change is shoved on us in professional development, there's a "pop" of implementation, and then it mostly fades away.

Loveless doesn't try to explain this, but I'll go ahead and give it a shot. Every teacher is a researcher and every classroom is a laboratory. And every instructional technique, whether it's in my textbook or pushed on me by edict or sold to me in PD or is the product of my own personal research and development efforts-- every one of those techniques is subject to the same rigorous testing and data-driven evaluation.

Does it work in my classroom?

I can find you numerous elementary teachers who took their newly purchased Common Core math textbooks, tried the recycled New Math instructional methods and pacing in the texts (because most of us will try anything once or twice) and then said, "Well, my students are confused and can't do the work. So I will now add a few days to the suggested pace of the book, and I will teach them how to do this The Old Way so that they can actually get a handle on it." The "fading novelty" looks a lot like "adapting or rejecting new ideas based on the real data of the classroom." And since Common Core's novelty is the product of well-connected amateurs and their personal ideas about how school should work, the novelty has indeed faded swiftly.

To the extent that we are allowed to (and that is the huge huge huge problem facing teachers in some districts-- they are no longer allowed to exercise their professional judgment), we do what meets the needs of our students. We do what works. We don't stick with something that doesn't work just because some textbook sales rep in a PD session or some faceless bureaucratic ed amateur in an office said we should stick with it.

Loveless suggests there are two plausible hypotheses. 1) As educators get better at using CCSS techniques, results will improve. 2) CCSS has already shown all the positive effects it ever will. I'm going to say that both are correct, as long as we understand that "get better at using CCSS" means " steadily edit, revise, change and throw out pieces of the Core based on our own research and knowledge of best practices."

Loveless does highlight one measurable effect of the Common Core-- the increased emphasis on non-fiction and the concurrent de-emphasis on fiction. He has data to back this up. And he also knows what the shift really means:

Unfortunately, as Mark Bauerlein and Sandra Stotsky have pointed out, there is scant evidence that such a shift improves children’s reading.

He also notes that more non-fiction doesn't necessarily mean higher quality texts, noting that two CCSS supporting groups provide completely different ideas about curriculum.

Loveless notes that analysts tend to focus on formal channels of implementation and ignore the informal ones. It's a good catch. A top-down directive from a state department of education can carry much less power than teachers sharing the video clip of CCSS architect David Coleman explaining that "nobody gives a shit what you think or feel." And politics are involved in the Core (and always have been, since the Core was imposed by political means).

Finally, he notes that implementing top-down curriculum and instruction reforms always runs afoul of what transmitters think, and boy, do I agree with him on this one. Every top-down reform is like a game of telephone, and each person who passes the program along reads into it what they personally think should happen.

As the feds tell state departments of ed and the department tells its functionaries and they tell their training division and they tell superintendents and superintendents tell principals and principals hire professional development ronin-- at each handoff of the baton, someone is free to see what they believe the program "must" require.

Loveless uses the example of non-fiction reading, postulating that an administrator who had always wanted to dump fiction for non-fiction would be given protective cover by CCSS. But that rests on a pretty explicit reading of what CCSS says about itself. This sort of top-down implementation also gives rise to policies that involve reading between the lines, such as an administrator who wants English teachers to teach less grammar and uses Common Core as justification. And of course the standards have been completely rewritten by test manufacturers, who interpret some standards and leave others out entirely.

In fact, some folks make a curriculum argument based on what the standards don't say at all. The rich content crowd insists that implementing Common Core must involve rich, complex texts from the canon of Important Stuff, and their argument basically is that because Common Core doesn't really require rich content because otherwise, it would just be a stupid set of bad standards emphasizing "skills" while leaving a giant pedagogical hole in its heart where the richness of literature should be. They must mean for us to fill in the gaps with rich text, they argue. Because surely the standards couldn't be that stupid and empty. (Spoiler alert: yes, they are).

In fact, in an otherwise pretty thorough brief, Loveless misses another possibility-- the Common Core Standards are limited in their ability to influence instruction because they aren't very good. Can I influence the work of cabinet-makers by putting bananas in their tool boxes? Can I influence how surgery is performed by telling surgeons to wear fuzzy slippers into the operating room? The implementation problem remains unchanged-- it's impossible to have a good implementation of a bad program.

58 comments:

Oh if only it HADNT made its way to the classrooms... I can state definitively, armed with 5 years of class work from 3 kids collectively spanning 10 grade levels (grades k-9) , that common core is alive and well in Tennessee classrooms. It is faithfully implemented daily and successfully causing it's inevitable confusion and chaos,turning kids off to both math and reading. They are turned off to math because (at the lower levels especially) it seeks to teach the "why" long before teaching the "what". It is truly the definition of illogical and kids as young as 1st grade are very articulate about being a combination of frustrated and insulted. Parents everywhere know exactly what I mean. Why are they turned off to reading? because adhering to the standards as written replaced inspiring, well written stories with dry, bland, poorly written materials that often are only excerpts vs. entire stories. They are turned off to reading because the stories are gone. Gone. Intentionally. I too believed at first that the struggle was with implementation. That the failure to find aligned materials led to this endless stream of chaos, stupid and insulting approach to math and bad, boring "texts" vs. stories and books. But 5 years in Its obvious I am wrong. The chaos isn't from a failure to properly implement. It is the result of correctly implementing. Common core, properly realized, implemented and taught is, at its core, chaotic, disconnected and backwards. At its core, the CC relies on that chaos, combined with a deep level of developmentally innaprioprite material to appear "rigorous". The corresponding tests rely on a combination of above-grade-level content and difficult to navigate software to appear "rigorous".There is very little about about the CC and its tests that is authentically "challenging" vs. fundamentally "bad". Unless you count expecting kids to be taught above their developmental capabilities "rigorous". Unless you count the skills it requires to operate badly designed computer software as "raising the bar". There is no "there" there. What kids learn during this decade of hell ieill be in SPITE of the CC - or by accident, or to the credit of teachers who look for "work arounds." This thanksgiving those teachers are at the top of my list of blessings. Without them- all would be lost. And I mean that. (Which may prove your original point..) So... Yes. CC made it to the classroom. Now it needs to leave.

I think the CCSS Math standards are a good step away from teaching computation and toward teaching math. You might want to look at this post about the (internet) famous CC math check: http://www.patheos.com/blogs/friendlyatheist/2015/09/21/the-dad-who-wrote-a-check-using-common-core-math-doesnt-know-what-hes-talking-about/

With all due respect, I do not need to go to the Internet to form my opinions of CC math. I have seen it in action from k-9th grade, up close and personal, at my own dining room table, . It is backwards. It teaches "why" before "what". It goes against an accepted axiom of mathematics: simple is better than complex. Take the shortest route. My reasons go on and on. And I am not alone in my analysis.

One point worth mentioning. Typically those who like CC math like to explain the methods. I get the methods. I can do CC math. But here's the thing: doing the methods actually require the very things CC says are useless: Math facts ! Like..quickly adding 2 numbers, quickly multiplying.. Etc. Usually the explanation goes something like this: Take 12 + 8. you just break the number down to a ten and give what's left to the other number. So... 10+ 10STOP! How did you do that? You quickly subtracted 12-10 ... (Which according to CC is a useless math fact) Then you quickly added 8+ 2. (Another allegedly useless math fact) .... I'll stop there. I've made the point. CC methods ASSUME KNOWLEGE. The very knowledge the CC derides. Math facts. Memorizing math facts. You CANNOT play around with "why" if you do not first know "what". . Adults who like these methods like them because they are not 8 years old and lost at the very sentence "oh just turn that 12 into a 10". If we had taught them their MATH FACTS- they could play all day with sticks and circles and critically think about WHY. But cc calls rote memorization useless. Ha! Do you know what our 3rd grade teachers tell us? Please oh please drill them on their times tables before sending them here! CC math is a set of nifty little tricks that' many of us adults use AFTER we've learned our math facts.

I am living the pre Algebra, Algebra hell right now with Common Core. Noticed that the 7th grader last year wasn't bringing home much skill work. She was getting 'B's. I took her for independent testing after 3rd quarter and low and behold, she had no Algebra skills (scored a 40% on an assessment). So now she sits in Geometry (which is another joke) as an 8th grader and I PAY for her to be tutored in Algebra. I just had to pull my son out of pre-Algebra and put him in regular math because he couldn't do the work. In fact, the 8th grader in Algebra tutoring couldn't do his homework.I thought CC was supposed to go into fewer topics but in more depth. That's just BS. They are flying through all of the basic skills needed for higher math so that they can teach what will be on the stupid test? It also seems that the Algebra I is being taught in pre-algebra, so I don't know what will be taught in Algebra II (probably pre-calc)? If you don't spend time having these kids learn the basics, they will never be able to take any kind of higher math in HS or College. The College remedial math programs are already overcrowded. I hate to see what it will be like in a few more years if this keeps up. I'M FRUSTRATED with the whole system, but the math standards stink the worst!

I knew from the beginning that you had formed an opinion, my hope was that you might be willing to consider the possibility that your opinion was incorrect.

Here is another paper you might read: Lockhart's Lament. Written in 2002 by a Ph.D. Mathematician who was teaching K-12 mathematics at a NYC private school (Link: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf)

edblisa,

The basic skill you need for higher math is logical thinking. This is one of the tragedies about the traditional way mathematics is taught. Set theory is far far more important to higher level mathematics than memorizing how to do integration by parts

Cripes you're an arrogant prick, you know that TE? Did you read Frustrated's first paragraph about the wealth of EXPERIENCE she's had with CC? Three kids over five years spanning ten grades? Did you get that part? And you think your mansplaining is going to convince her otherwise? If you're wondering why people don't seem to like you (assuming you have the self-reflective capacity to wonder such things), this would be a good place to start.

sorry teachingeconomist....you're nothing but a troll. If you are having a house built and your foundation isn't built well, guess what happens to your house.....it falls apart. Same with math skills. If the basics aren't well learned, there is no learning higher math. Math builds from the bottom up. People like you get a rise out of feeling smug and hiding behind the anonymity of a computer screen.

Did you try reading the mathematician's lament? The skills that you speak of ARE NOT the skills of mathematics, but the skills of calculation in one of many ways to look at arithmetic. The bottom of math is logical thinking, the ability to understand that a p-adic number system is just another way to extend the integer system, not something strange.

Talk to some research mathematicians about what mathematics is and what is taught in K-12. They will tell you there is little overlap.

Want me to tall you about my experience with the non-CCSS math education in my local public schools? The only thing that saved my older son's interest in mathematics was getting him into the hands of research mathematicians at our local university as quickly as possible.

Did you read the Lockhart's Lament? It would be a better use of your time than thinking up novel ways to personally insult me.

You have to copy the link into your browser as the blog does not seem to allow live links. Alternatively you can simply search for Lockhart's Lament. You should find a pdf file on the Mathematical Association of America. There is also a book that he wrote, so be sure not to click on an Amazon link.

The teachingeconomist is probably a "true" math person. He sees the world as nothing but math. I grew up in a household with one of these high IQ, Mensa,math people. It was difficult for my parents and it was difficult for us siblings to get along with this family member. I refer to him as a freak of nature. Yes, he was very intelligent, but he had no social skills. Where the good Lord giveth he taketh away. I feel sorry for people the likes of teachingeconomist. They don't get along with family and friends are hard to come by. The internet is the only place where they can feel like they get along :(

Once again, rather than simply making denigrating assumptions about me, why not actually read Lockhart's Lament? I know that reading it will take a good deal of more effort than simply constructing an image of me, but often learning does require effort.

You're not getting it, TE. Your experience has absolutely NOTHING to do with Frustrated's experience. What you're doing is called mansplaining - when men (predominantly, although women can be guilty too) think that their head-knowledge outweighs the actual, lived experience of, usually, a woman. If Frustrated found God Himself extolling CC in the Bible, that would not change the experience that she has actually had with her real, live, flesh-and-blood children. I know you're not an actual, real live flesh-and-blood person, so I know you don't get it, but maybe sometimes you'd just be better of if you'd STFU.

And apologies to Peter for such an outburst on Thanksgiving Eve, but TE has long had it coming.

i tried to read both links. Neither work- even when pasting the link into browser. Teaching methods aside, (which is a big aside) The major flaw with CC math/standards is that it/they are developmentally absurd. I am sure you speak correctly when you say "higher thinking" is the skill needed for mathematics. But all in due time. And k-3 is NOT in due time. The 4-6/7 cirriculum? Not in due time. I've just started the HS experience so Im holding my thoughts to see what comes.

DEVELOPMENTAL STAGES; 2-7 years ( through 2nd grade) is the pre-operational stage. A short list of abilities include :Concepts formed are crude and irreversible. Easily believe in magical increase/decrease Reality not firm and yet...By second grade we are teaching these kids WHY math is what it is. It's gobbeldy goop to their literal minds. And a total waste of what they ARE wired for. Eating up facts! Absorbing them like sponges! You can teach a 2nd grader , well....ANYTHING YOU WANT! They love lists and facts and dates and events. An especially advanced second grader could learn multiplication tables. And love it! But they can't look at an "x" and understand it's a variable. Variables don't exsit in their world unless it's something they themselves made up.

Age 7-12 is the concrete operational stage ( through grade 6 or 7)

I won't list it out , if you're interested, google it. CC is as badly matched to this stage as the previous. Maybe a teensy tiny bit less badly matched, What a grand endorsement.

The point is...12 and up (6th to 7th grade and beyond) begins the formal operations phase. It is here- and not UNTIL HERE - that the typical child can function in abstract thought. Armed with facts, they can begin to dive deep,hard and shockingly fast into the "why" of.... Well, everything! This is the critical thinking stage. This is the part where "what" (unless you haven't actually learned "what" .....which is how we end up with even high school being "lost"...) reveals its delicious mysteries as to "why" it is what it is. The Magic a second grader found in aquiring facts and assembling lists, now pays off as another magical time when those facts and lists are USED AS TOOLS- they are the clay they hold as they manipulate it with their thoughts and their exciting new ability to analyze and critically examine. Lost. All lost. Because it's all out of order. Because we "raise the bar" to "challenge" them to do at 9- or even 7 or 8- what they can't do until 12. And at the same time, we squander what they CAN do at 9 because facts/memorization aren't "higher order skills".

Can't re-wire the human brain. And I can't figure out why in the world we would want to. Learning can be a magical thing. I so loved school, so very very much. But I was allowed to function at my developmental level. This generation is not. Though there are many other problems with common core, that it is developmentally incorrect should be the only problem needed to bid it farewell. And that, sir, is the beginning, the middle,and the end of my argument as to why it is nakedly obvious to anyone who bothers to look that education is fundamentally off track. If you disagree, we can agree to disagree. But no, I will not be convinced by a theory or an argument or a study or a response to an angry father who wrote a silly check out of frustration. Mathematical theories may have something to teach me. But honoring a child's development has something to teach ANY theory that discards it. CC math discards it. To the detriment of us all. Again- HUGE, GRATEFUL, SLOPPY,LOVE FILLED HUGS AND KISSES TO THE THOUSANDS AND THOUSANDS OF TEACHERS PLANTED FIRMLY BETWEEN ALL OF OUR KIDS AND THIS DISASTER. I'll stop shouting.. But you are the safety nets that save these little souls. Dramatic? I really don't think so,

I am not sure why the link will not work, but just search for "Lockhart's Lament" and "common core math check". The links should be the first or second result.

Here is a small part of Lockharts's Lament:

"The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such. Everyone understands that poets, painters, and musicians create works of art, and are expressing themselves in word, image, and sound. In fact, our society is rather generous when it comes to creative expression; architects, chefs, and even television directors are considered to be working artists. So why not mathematicians?

Part of the problem is that nobody has the faintest idea what it is that mathematicians do. The common perception seems to be that mathematicians are somehow connected with science— perhaps they help the scientists with their formulas, or feed big numbers into computers for some reason or other. There is no question that if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category.

Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.

So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G.H. Hardy’s excellent description:

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

I urge you to read the whole essay.

Let me close by quoting something my middle child stated just after graduating from high school about his frustration with the traditional non-CCSS math curriculum he faced:

I don't mean to say that mathematics is something most people cannot grasp, but simply something they were never taught. If I seem angry, it is because the material that passes for math in public schools nearly turned me off mathematics forever, and had it done so I would never have realized the sublime beauty of the subject and never felt the peace and joy that has come with understanding it.

I can assure you that despite Dienne's statements to the contrary, my children are flesh and blood.

*Pulls hair in utterly baffled frustration* TE, you do realize that YOU are the one who said that Frustrated's opinion is "wrong", no? YOU are the one trying to mainsplain to her. I have not read anywhere here where she has told you that your opinion is wrong. Even a robot like you should understand the difference.

I, too, was very frustrated by the Common Core inspired math curriculum in North Carolina. My son was in a Math II course last year. North Carolina decided to go with integrated math and mix geometry, algebra, and trigonometry in a series of three courses: Math I, Math II, and Math III. So far, it has been a disaster. The Math II course covered so many topics at such a fast pace that only the best students could keep up. Spending just one day on a difficult topic like rational equations before moving on to another topic that is not really related the next day makes no sense. This happened every week. I had to tutor my son constantly, and he still struggled in the course despite being good (not great) at math. He and many of the other students would have been better off with the old curriculum and its sustained focus on topics.

Also, I hate that Common Core has basically gutted geometry. My son didn't do any proofs and he won't do any in high school. Exposure to the logical thinking required by doing geometry proofs is good, and Common Core has pretty much eliminated that from high school.

Art, cooking, painting, music etc. begin with basic facts too. People misunderstand art as a free form flow of ideas disconnected from rules. That's not what art is. Yes- math is a pattern. So is art. To become a master at anything you need to learn the language. Math is no different. To see the pattern emerge you must be able to see that emerging pattern first. Which is through arithmetic. Perhaps a genius can see this sooner than typical humans. But typical humans need to learn how to read music before they can compose music. Typical children need to know 8x3 is 24 so that they can decompose those numbers to understand why. None of what you are saying changes the fact that "what" comes before "why". Perhaps your children did not start their math education with CC at the lower ages? Perhaps they already were armed with the information before they began to deconstruct the "why"?

You are correct that I disagree with Frustrated mom. I stated why I disagree (because I believe she is mistaken about what is fundamental to mathematics) and I give links to more extensive arguments to support my view, one written by a National Board Certified Teacher and another written by a research mathematician who had turned to teaching K-12 mathematics that was posted by the Mathematical Association of America (and later expanded and published as a book).

Your response was to call me names and suggest that I am not a human being.

Eric,

It sounds like your local school/district might not be implementing the core very well. I do not know where you live, but I often look to Kentucky for examples of reasonable thinking about implementing the CCSS. Here, for example, is a link to the Kentucky implantation of the CCSS for geometry: http://education.ky.gov/curriculum/conpro/Math/Documents/High%20School%20Geometry.pdf . If you scroll through it you will find it is littered with proofs. Here is my favorite statement from geometry unit 1, G.CO.9:

Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.

Frustrated,

My students were never taught the "why" in K-12 math, though they often asked about it. That is why we started sending our middle child to the local university for his mathematics education.

Did you read Lochhart's Lament? It begins with a musician's terrible nightmare: a group in charge of musical education determines that

Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.

First, I live in NC, and the local district did not come up with the new Common Core inspired curriculum. This was done at the state level. So, the entire state threw out a better high school math curriculum for a new one that was somehow consistent with Common Core. Recently, the NC legislature passed a bill to re-examine Common Core. A committee is working. They will probably recommend a move away from the Common Core standards.

Second, I see no reason that states, who are spending most of the money on K-12, should adopt national standards, especially when the standards are so flexible that they can be implemented in hundreds of ways. It seems to me that the implementation problem is really caused by the vague nature of the standards. It was perfectly foreseeable that many states, if not all, would not implement such standards very well. That seems to be what has happened.

At this point, I'm all for NC ditching the Common Core standards and going back to what it had before. My 9th grader did not learn as much math as he should have last year because of this grandiose experiment that has largely been a failure. This is what has made me very angry about the entire Common Core movement. The math courses that I had thirty years ago in high school were so much better the nonsense that my son had to suffer through. Basically, if NC had not adopted Common Core, then my son would have had a better math experience last year. That's the bottom line for me. I don't really care if this is an implementation problem or not.

I don't find the math standards to be particularly vague. Here is a link to the math standards for geometry, for example: http://www.corestandards.org/Math/Content/HSG/introduction/

The problem with the way it was before is that the traditional mathematics curriculum sets up so many obstacles to students learning actual mathematics that only extremely talented and determined students every get to actually do math. Remove the obstacles to doing math and more students will succeed.

I don't really care what is happening in Kentucky. I care what is happening in my children's schools in North Carolina. And in NC, Common Core has been a disaster for high school math.

As for your idea of actual mathematics, whatever that is supposed to mean, I don't really care either. I have a son who is interested in majoring in science in college. I would like him to leave high school knowing some basic knowledge of algebra, geometry, trig, and maybe calculus and stats. That's some of the stuff he'll need for biology, chemistry, and physics in college. So far, the Common Core inspired curriculum isn't giving him much of what he needs. For NC high school students, the move to Common Core created more obstacles in math instruction, not fewer. To me, that's a huge problem.

teachingeconomist, I was familiar with the "Lament." And he makes a good point, which - for ed reform purposes - raises a couple of questions:

1. What's the point of teaching math - to inspire children with its beauty, or to teach facts and operations which will be required in various different specializations?

2. What do we mean by the "basics?" Math facts and commonly-used operations & formulae, or logical thinking?

My field (composition) raises the same questions: why do we teach students to write? And where should we start?

Both good questions. But here's what Frustrated is getting at: CCSS tries to address this by bringing in all this elaborate metacognitive stuff, which is completely meaningless to K-3. Small children DON'T DO abstractions. Period. My daughter, aged 6, was bringing home math homework that always ended with some version of "Why is this the right answer?" to which she generally wrote, "Because it's what you were teaching us to do." Natural math enthusiasts might find this an exciting challenge, but to most children, including perfectly intelligent ones, that question is perplexing, like asking them why they wear pants, or why they eat lunch at 11:30 instead of 12.

Abstractions only make sense when they arise out of non-abstract knowledge. Ideas about revolution, or animal husbandry, or indeed, school reform, take shape out of a thousand facts at our command. Where we try to trade in abstractions that don't correspond to realities, we start bullshitting, which is what a lot of kids are learning to do in the early years.

TE, I read Lockhart's piece and it was really interesting. I totally agree with him about how bad math is taught now. My experience with it was, as he says, with "no historical perspective or thematic coherence, a fragmented collection of assorted topics and techniques" and "mindless regurgitation of formulas" and "contrived exercises".

I understand what he's saying as far as that the way it's taught, all the joy is taken out of it because what he sees as real math is the underlying conceptual framework, and that its ideas are a philosophical art of logic and symmetry. I've seen other math teachers wondering why it was so important to memorize the quadratic formula, and I myself was always more interested in wanting to know how somebody derived it than in plugging numbers into it, which is boring. I did see an art and symmetry in similar triangles, and in how cool it is to be able to solve an equation by doing the same thing to each side, and it's still equal. And I agree that insisting on terminology like (latin) "quadrilateral" instead of (anglo-saxon)"four-sided shape" is distracting, the same way I think it's distracting in grammar to use terminology like "preposition" instead of "little connector word".

However, I seriously doubt Common Core was written by a mathematician like Lockhart. In fact, I think Peter would agree with Lockhart that there should be "no standards, and no curriculum. Just individuals doing what they think best for their students" and that people having the same exact basic knowledge or being measured to find out their "relative worth" is not important. He seems to disagree with you, TE, about math being important because it's "the language of science"; he's such a "pure mathematician", he thinks it being "useful" is just a by-product of the art.

I agree with Lockhart that the most important thing to be a good teacher is the teacher's enthusiasm, love, and deep conceptual knowledge of their subject, but I don't agree it's the only thing. He says they need no lesson plans, methods, tools, or training, and is very dismissive about studying early childhood development and "whatnot". Lockhart obviously has a natural, intuitive understanding of math that most people don't, and taught at a private school, probably with very small classes.

To be their most effective, most teachers need to understand cognitive learning theory: how people learn, what types of strategies might work with most kids, what different learning strategies work with different learning styles. Lockhart, like many teachers who don't know enough cognitive theory, believes everyone learns the same way he does. (He also has the advantage of being one of the few math people who are also good with words; in fact, poetic.) He's of the opinion that the best way to teach is to give students a problem and let them struggle and get frustrated. That can work with students who have an intuitive understanding of the subject, but not most. Most also need more fundamentals than he advocates.

I know that frustrating me by giving me a math puzzle would just make me give up. I'm not good with puzzles. What would help me would be to show me how Euclid figured it out, but there's no way I'd get there on my own. I've often thought the only way I could understand math would be to teach me "the history of mankind's relationship with numbers". That would be fascinating to me, but might not be interesting to someone else.

The same way Lockhart says "Appreciation of poetry doesn't come from memorizing a bunch of poems, it comes from writing your own." No, not for me. The only way I would ever be able to write a poem would be to follow a formula, like one noun, then a line with two adjectives, then three adverbs, then two verbs. And the result would be pure crap. Whereas some people write poetry from an early age without studying it at all and maybe not even reading that much of it, and it's good. We all have different talents. I finally got to where I could appreciate some kinds of poetry, and I love reading plays and novels, but I'll never be able to write one. Some mathematicians think you should study geometric proofs to develop logical deductive thinking. I naturally think analytically, deductively, and logically, but not for doing mathematical proofs, I just have no interest in the topics. However, I find the philosopher and mathematician Spinoza's use of axioms and proofs to try to prove his ideas fascinating. I think some of his axioms are silly, but according to Lockhart, I'm allowed to.

It's also true, as Frustrated mom says, that developmental readiness is key, and that seems to be a problem in both the math and English Common Core standards. As she says, capacity for abstract thought starts developing for most kids around the age of 12, and I know that's when mine did. When I tried to help my son, at the age of 14, try to undertand the algebraic concept of a letter standing for a number, his response was like Chris Farley's in the movie Almost Heros when Matthew Perry was trying to teach him to read and tried to show him that "A" and "a" stand for the same thing: he felt like his head was going to explode. So that's why cognitive theory is important. And there are some kids who have the type of personality that they're fascinated by formulas, others who are fascinated by application to science, some like me who appreciate historical perspective, and some who are fascinated by the pure art of math. But I do agree with Lockhart that the goal of teaching math should be to "allow access to beautiful and meaningful ideas."

Well said. And I'd like to add a defense pf memorizing poems, btw, much reviled by lit lovers, who think it's pointless. I disagree. Once memorized, those words are yours; and one day you find them bubbling up to give shape to a feeling. It's one of the best ways to learn the love of literature.

I think your posts fit well together. Eric wants his son to know "the basics" of algebra, and Madeleine asks what do we mean by the basics. I think the most basic question about algebra is what is an algebra and how many algebras exist? Eric, I would be surprised if your K-12 mathematical education 30 years ago ever discussed the definition of an algebra or even mentioned that different algebras existed. If you would like a list of the different types of algebras, here is a link to a paper listing them: http://arxiv.org/pdf/1101.0267.pdf

Madeleine,

I disagree that young children can not make abstractions. At a very young age my children could identify a never before seen cat as a cat, a never before seen before tree as a tree, a never before seen person as a person. How can they do that without the abstract idea of a cat is, a tree is, a person? Can your six year old not identify a dog correctly, though she has never seen that dog or even that breed before?

The right mix is a good question, but the way we traditionally teach math is not a mix at all. Think about the assignments that might be given in an ELA class. A student might well write a pathbreaking poem, a short story that would win the O. Henry prize, or the finest analysis of a Shakespeare play ever written. That will not, can not, happen in response to any assignment given in a K-12 math class. If a student does do something novel, does construct a novel way to analyze the assigned problem, would the teacher even recognize it or would the teacher count off for not following instructions?

Rebecca,

As I said above, I think that young children make abstractions all the time. They have the idea of object permanence, that is that objects exist even when they are not observed. As i said above the idea of a tree is an abstraction, the idea of a dog is an abstraction. I am very sure that you made those abstractions well before you were 12.

That's recognizing similarities and categorizing, which is different. That children can do at least as soon as they can talk, because at its most basic, that's what nouns are; "naming" or labeling is putting things in categories. I'm talking about a symbol representing something that's entirely different, like a letter representing a number; I'm not talking about recognizing the number 2 written in different handwritings, which would be analogous to the cat. There's TONS of research on this, TE. The first time I remember using abstract thought was in Sunday school in seventh grade. The teacher asked how it could be that the Bible said God created the earth in 7 days, and I said maybe it didn't mean a literal day. So we're talking about figurative thinking, not recognizing concrete objects that are similar enough to be put in the same category. Though ages for child development are never written in stone; there are parameters of variance, so to me there's nothing wrong with introducing topics just to see if somebody gets it, but you have to be careful about trying to make them responsible for understanding stuff too soon.

Are you actually suggesting the students be taught the philosophy of mathematics in high school? That is not a good idea. I just want my son to be able to work with a variety of equations and problems. He is not ready for philosophical discussions about the different kinds of algebra, and certainly not as a 9th grader. The philosophy of mathematics can wait until college or graduate school.

If students need to know the philosophical foundations of algebra before doing any algebra, then wouldn't students need to know the philosophical foundations of numbers before they learn to add and subtract? Let's start with the most fundamental question: what is a number? This approach is not going to work well in elementary school.

Yes -- as Rebecca says, obviously, even small children can do certain kind of abstraction. Toddlers have object permanence; pre-schoolers, mostly, have "theory of mind."But they can't *analyze* or even explain either ability. Try asking a 3 year old why he knows that an errant ball still exists behind the couch. Or try to get the 5 year old to explain why it's significant that the cat does *not* appear to know that the ball is behind the couch. These two conversations will go nowhere.That is the kind of abstraction we are talking about: the ability to talk analytically, to draw and explain inferences, to make reasoning transparent. Because CCSS wants "critical thinking" to be testable, students find themsellves badgered for explanations which can't possibly have meaning.which is not to say, again, that "Lament" doesn't have a good point. There are certainly better and more creative ways to introduce children to math. But I don't think CCSS is one of them.

You might want to take a look at the link I posted to Rebecca. Let me quote the closing paragraph:

In fact, I’d suggest that complete mastery of concept across materials, types of query, and times is a good indication that the concept was introduced at a developmentally inappropriate time. We waited too long–the child probably already knew the concept.

@Rebecca - *exactly.* There's much to be said for *introducing* these concepts. Ask the questions! At any age! Actually, my example about the cat and the couch stemmed from a conversation I had with my daughter when she was little. We talked about lots of things: the difference between really meaning things, and not really meaning things (ie, literal v. figurative language); about the complex nature of the self (when you are really mad, don't you also see yourself being mad, and think that you look funny?) - all kinds of stuff.

But this is NOT what CCSS is about. CCSS defines intellectual achievements which must be demonstrated, through some form of self-aware self-reporting, in a standardized test, every freaking year, by every child in the public education system - and in many places, which will be used to evaluate the teacher's efficiency.

This is not even remotely what Willingham is talking about (I've read lots of pieces by him and have yet to disagree with a word), or Lockhart, or any of those many people who have long complained that the field of education is rife with dodgy and poorly-thought out theories.

Here is your earlier question: "I think the most basic question about algebra is what is an algebra and how many algebras exist?" These questions are philosophical in nature. I took it that you thought these questions should be addressed in high school math classes since you were responding to my criticism of high school math curriculum. I then extended this logic to arithmetic. Certainly, the most basic question about numbers is: What is a number? Again, this is a philosophical question.

If you want to have a discussion about the true nature of mathematics and mathematical objects, then I suggest you drop by the philosophy department at your university.

It would seem that you have come a bit off your position that "Small children DON'T DO abstractions. Period. (Your November 26, 2015 at 3:58 PM post, capitalization yours)". I agree with your new position that "...obviously, even small children can do certain kind of abstraction. (Your November 26, 2015 at 3:58 PM post)" The very interesting question, still the object of research, is which kind of abstractions and how large a variance there is across individuals. I think it is likely that children are more capable of mathematical abstractions than you believe.

Rebecca,

Nowhere have I suggested that small children be made accountable for anything, including asking them to verbalize to someone else how they were applying an abstract idea.

Eric,

The question about how many algebras exist is not a philosophical one, but an empirical one. The way K-12 traditionally teaches this is like teaching students there is only a single language in the world. While what constitutes a language might well be seen as a philosophical question (indeed there is a field in philosophy called philosophy of language), the existence of French and usefulness of Mandarin is not a philosophical question.

Take modular arithmetic. If you can read an analog clock you can understand the basic idea of it.

Here is a fun post about it: http://betterexplained.com/articles/fun-with-modular-arithmetic/

If your son is interested in internet security and cryptography, modular arithmetic will be a necessary tool in his tool chest.

And not just being understood, understanding also. Lack of reading comprehension skills, lack of logical deduction skills, lack of skill in making inferences, lack of skill in seeing and understanding the main idea...What do you think the BS tests are?

It wasn't my post, it was Frustrated mom's, and I told you: they're held accountable in the state-mandated 3rd grade tests (BS "Big Stakes" Tests). In some places they don't pass to fourth grade if they don't do well on them, and even if it doesn't count for that, they feel like failures if they don't do well. Plus they're trying to train them in this inappropriate stuff from kindergarten and it all probably counts in their grades. I'm sure Frustrated mom could explain this better but she's probably tired of trying to talk to someone who seems so dense, and not everyone is as lucky as I am to be retired and have time I can afford to waste. It seems self-evident that they're held accountable by the tests and in any case I don't know why you can't just take her word for it. What is wrong with you? I've successfully taught foreign language to kids who were cognitively disabled, and they certainly understood things much better than you do. Your lack of understanding is so incredible I think you must be just playing dumb, but I can't imagine why anyone would want to pretend to be so stupid.

The third grade tests have nothing to do with the CCSS. They are tests of literacy. See this article: http://www.huffingtonpost.com/2014/09/30/third-grade-literacy-retention-laws_n_5907272.html

My young children at least did not get grades until junior high, and even those grades did not really have any consequences until high school. In high school the only tests of any real importance were teacher generated tests that determined grades and eligibility to graduate from high school.

The major problem is that what you think is self evident in fact does not happen. Many people "know" that there is no global warming, that there were thousands tailgating to celebrate the destruction of the twin towers, and that Obama is a Muslim born in Kenya. Should I just take those peoples word that these things are true or should I actually look at the evidence?

What blew me away in this whole process was the way the first transmitter in the process -- Coleman -- disregarded the text of the standards. Focusing on the whole non-fiction/fiction issue was CRAZY. It has no business being in the standards at all (and it isn't, except the introduction), it has never been explained sufficiently to even accurately implement it across a school's curriculum, etc., etc. It is wacky on every level. That entire subject is off-topic. Coleman already talked like the 10th person in the telephone tree, not the first.

we supervise student teachers (my colleagues) and when you walk in to observe a 2nd grade classroom, for example, the student teacher has to conform to the "curriculum" that has been adopted in that school; where it is the "common-ness of core" it is frequently an inappropriate lesson way above the actual development of the child -- so as a supervisor what can you say? "I know you are required to teach this [for example spelling lesson] but do you see how it is not appropriate for the children you have in this group?" And call their attention to it… as an aside, one of my colleagues has a grandson whose parents retained him a year in kindergarten rather than sending him on knowing that the common-ness of core would be inappropriate for him in first grade and hoping that one more year the rigid application of the "common ness of core" will be alleviated. Here is hoping.

I am agreeing with Tom Hoffman; we at one time had a broader definition and more comprehensive view of what is taught in "reading"… here is an example from Keith Stanovich who cites "research studies of printed texts ranging from children's stories to adult books, from comic books to popular magazines, showing that each type of reading material contains a far greater number of rare words than TV shows or adult speech (even if the adults are college educated)…. furthermore, the student learns more vocabulary from reading juvenile fiction than from watching prime time TV." Coleman et al have taken a very reductionistic view of "reading" in order to get items that conform with the "bubble in" tests that the computer spits out so that the governor can have "data" and the real estate moguls can sell more property in the most affluent districts in our state.

In my town at least, far from either coast, every real estate advertisement lists the elementary, middle, and of course high school district. It is pretty clear that people are buying access to schools.